Lemma 37.3:

For a pair $(X, A)$ of spaces the connecting homomorphism $\delta_n : H_n(X, A) \longrightarrow H_{n-1}(A)$ is given by

$$\delta_n\overline{c} = \overline{\partial _nc},\quad c \in Z_n(X, A). \eqno(37.4)$$

Despite the notation, $\partial _nc$ in (37.4) is not a boundary in $S_{n-1}(A)$ since $c$ is not a chain in $S_n(A)$ but a chain in $S_n(X)$. If $\zeta$ is a cycle in $A$ then for sure, it is a cycle in $X$ as well but then it may be actually be a boundary $X$, in other words $H_n(i)\overline\zeta = 0$. This happens precisely when $\overline\zeta$ is in the image of $\delta_{n+1}$ by exactness of (37.3). Figure below depicts a cycle in $A$ (annulus) which is a boundary in $X$ (the polygonal region). in The long exact sequence in the preceding theorem is natural in the following sense.